Chromatic polynomials of simplicial complexes

Transcription

1 Chromatic polynomials of simplicial complexes Two open problems Jesper Michael Møller Natalia Dobrinskaya Frank Lutz Gesche Nord Dietrich Notbohm University of Copenhagen Copenhagen May 31, 2013

2 Content 1 log-concave sequences and falling factorials 2 Colorings of simplicial complexes Chromatic numbers of simplicial complexes The chromatic polynomial Comparing chromatic polynomials of graphs and simplicial complexes 3 The d-chromatic lattice 4 Weighted colorings JM Møller (U of Copenhagen) Chromatic polynomials May 31, / 29

3 log-concave sequences Definition 1.1 (LC) A finite sequence a 1, a 2,..., a m of positive numbers is log-concave (LC) if a j 1 a j+1 a j a j for 1 < j < m. (a j ) m j=1 is log-concave a 1 a 2 a 2 a 3 a m 1 a m log a j (j, log aj) log a j 1 + log a j+1 2 log a j = (a j ) m j=1 is unimodal (j 1, log a j 1 ) (j + 1, log a j+1 ) (j, 1 2 (log a j 1 + log a j+1 )) j Example 1.2 Binomial sequence j ( ) m j is LC 1, 2, 5, 2, 1 is unimodal but not LC JM Møller (U of Copenhagen) Chromatic polynomials May 31, / 29

4 Falling factorials and Stirling numbers Two bases for the polynomial ring Z[r] j [r] j = r(r 1) (r j + 1), r j {}}{ = r r r }{{} j falling factorial base (FFB) monomial base (MOB) [r] 0, [r] 1, [r] 2, [r] 3,... base change r 0, r 1, r 2, r 3,... [r] m = m j=0 S 1(m, j)r j Stirling numbers 1st kind r m = m j=0 S 2(m, j)[r] j Stirling numbers 2nd kind S 2 (m, j) is the number of partitions of an m-set into j blocks [r] 1 = r 1 r 1 = [r] 1 j S 1 (m, j) is LC [r] 2 = r 1 + r 2 r 2 = [r] 1 + [r] 2 j S 2 (m, j) is LC [r] 3 = 2r 1 3r 2 + r 3 r 3 = [r] 1 + 3[r] 2 + [r] 3 JM Møller (U of Copenhagen) Chromatic polynomials May 31, / 29

5 Colorings of simplicial complexes Definition 2.1 (Colorings of simplicial complexes) A (weak) (r, d)-coloring of the simplicial complex K is a map col: F 0 (K ) {1, 2,..., r} such that col(σ) = 1 = dim σ < d for all simplices σ K. (K, d > 0.) (4, 1)-coloring (2, 2)-coloring (2, 3)-coloring JM Møller (U of Copenhagen) Chromatic polynomials May 31, / 29

6 Chromatic numbers of simplicial complexes Definition 2.3 (The d-chromatic number of a simplicial complex K ) The d-chromatic number of K, chr(k, d), is the minimal r so that K admits an (r, d)-coloring. F 0 (K ) chr(k, 1) chr(k, 2) chr(k, dim K ) 1 Example 2.4 (Do we know the chromatic numbers of any complexes?) K = D[4] (2, 2)-coloring chr(k, 1) = 4 chr(k, 2) = 2 chr(k, 3) = 2 chr(d[m], d) = m d JM Møller (U of Copenhagen) Chromatic polynomials May 31, / 29

7 Chromatic numbers of triangulable manifolds Definition 2.5 (The d-chromatic number of a compact manifold M) chr(m, d) = sup{chr(k, d) K triangulates M} chr(m, 1) chr(m, 2) chr(m, dim M) 1 Example 2.6 (Do we know the chromatic numbers of any manifolds?) K = S 2 chr(s 2, 2) chr(k, 2) = 2 Is there a triangulation K of S 2 with chr(k, 2) > 2? Theorem 2.7 (The 4-color theorem = chromatic numbers of S 2 ) chr(s 2, 1) = 4 and chr(s 2, 2) = 2 JM Møller (U of Copenhagen) Chromatic polynomials May 31, / 29

8 Chromatic numbers of S 3 Problem 1: What are the chromatic numbers of S 3? chr(s 3, 1) = FOR SURE chr(s 3, 2) = PRESUMABLY chr(s 3, 3) < UNKNOWN The standard triangulation K = D[5] of S 3 has chr(k, 3) = 2. There exists a triangulation K, f (K ) = (18, 143, 250, 125), of S 3 with 3-chromatic number chr(k, 3) = 3. Does there exist a triangulation K of S 3 with 3-chromatic number chr(k, 3) > 3? Theorem 2.8 (Chromatic numbers of spheres) chr(s d, d/2 ) = when d 3 PRESUMABLY JM Møller (U of Copenhagen) Chromatic polynomials May 31, / 29

9 Red-necked Grebe JM Møller (U of Copenhagen) Chromatic polynomials May 31, / 29

10 χ(k, r, d) is the number of (r, d)-colorings of K χ(k, r, 1) χ(k, r, 2) r r chr(k, 1) = 5 χ(k, 5, 1) = 120 chr(k, 2) = 2 χ(k, 2, 2) = 10 JM Møller (U of Copenhagen) Chromatic polynomials May 31, / 29

11 r χ(k, r, d) is the d-chromatic polynomial of K χ(k, r, 1) χ(k, r, 2) χ(k, r, 1) χ(k, r, 2) chr(k, 1) = 5 χ(k, 5, 1) = 120 r chr(k, 2) = 2 JM Møller (U of Copenhagen) Chromatic polynomials May 31, / 29 r

12 Simplicial Stirling numbers Compute the number χ(k, r, d) of (r, d)-colorings of K! Definition 2.9 (Simplical Stirling numbers) S(K, j, d) is the number of partitions of F 0 (K ) into j blocks containing only K -simplices of dimension < d. S(K, j, d) = S 2 (m, j) when K = K 1 K 2 = S(K 1, j, d) S(K 2, j, d) when F 0 (K 1 ) = F 0 (K 2 ) S 2 (m, j) S(K, j, d) S(D[m], j, d) with equality for j = F 0 (K ),..., F 0 (K ) d + 1, m = F 0 (K ) }{{} d S(K, F 0 (K ) d, d) = S 2 ( F 0 (K ), F 0 (K ) d) f d (K ) S(K, j, d) = 0 for 0 < j < chr(k, d) chr(k, d) = min{j S(K, j, d) > 0} JM Møller (U of Copenhagen) Chromatic polynomials May 31, / 29

13 Colorings and equivalence relations S(K, j, d)[r] j (r, d)-colorings of K col: K D[r] with col(f 0 (K )) = j [r] i -to-1 S(K, j, d) Partitions of F 0 (K ) into j blocks without d-simplices col: F 0 (K ) {1,..., r} {v F 0 (K ) col(v) = k} Theorem 2.10 (The d-chromatic polynomial of K ) The number of (r, d)-colorings of K is χ(k, r, d) = F 0 (K ) j=chr(k,d) S(K, j, s)[r] j JM Møller (U of Copenhagen) Chromatic polynomials May 31, / 29

14 Colorings and equivalence relations r possible colors r 1 possible colors r 2 possible colors 3 blocks with no d-simplices can be colored in [r] 3 ways from a palette of r colors χ(k, r, d) = F 0 (K ) j=chr(k,d) S(K, j, d)[r] j JM Møller (U of Copenhagen) Chromatic polynomials May 31, / 29

15 Colorings and equivalence relations χ(k, r, d) = F 0 (K ) j=chr(k,d) S(K, j, d)[r] j JM Møller (U of Copenhagen) Chromatic polynomials May 31, / 29

16 Colorings and equivalence relations χ(k, r, d) = F 0 (K ) j=chr(k,d) S(K, j, d)[r] j JM Møller (U of Copenhagen) Chromatic polynomials May 31, / 29

17 Colorings and equivalence relations χ(k, r, d) = F 0 (K ) j=chr(k,d) S(K, j, d)[r] j JM Møller (U of Copenhagen) Chromatic polynomials May 31, / 29

18 The two chromatic polynomials of a 2-complex χ(mb, r, 1) r 5 10r r 3 50r r 1 > > [r] 5 chr(mb, 1) = χ(mb, r, 2) r 5 5r 3 + 5r 2 r 1 > > 5[r] [r] [r] 4 + [r] 5 chr(mb, 2) = JM Møller (U of Copenhagen) Chromatic polynomials May 31, / 29

19 Chromatic polynomials of graphs Example 2.11 (Specialization to graphs) An (r, 1)-coloring of K is an r-coloring of the simple graph K 1, and the 1-chromatic number of K is the graph chromatic number of K 1. monomial basis (MOB) {}}{ F 0 (K ) b j r j = χ(k, r, 1) = MOB is ± j=1 MOB is LC falling factorial basis (FFB) {}}{ F 0 (K ) a j [r] j j=1 The MOB coefficients (b j ) alternate in sign The MOB coefficients ( b j ) are LC No roots < 0 χ(k, r, 1) has no roots < 0 m, m 1 vals χ(k, m, 1) > eχ(k, m 1, 1), m = F 0 (K ) FFB is LC Properties of 1-chromatic polynomials The FFB coefficients (a i ) are LC UNKNOWN JM Møller (U of Copenhagen) Chromatic polynomials May 31, / 29

20 Example of a 1-chromatic polynomial Properties of χ(og, r, 1) MOB is ± Yes MOB is LC Yes No roots < 0 Yes m, m 1 vals Yes FFB is LC Yes 1-chromatic polynomial in MOB and FFB χ(og, r, 1) = 64r r 2 137r r 4 12r 5 + r 6 = [r] 3 + 3[r] 4 + 3[r] 5 + [r] 6 JM Møller (U of Copenhagen) Chromatic polynomials May 31, / 29

21 Example of a 2-chromatic polynomial Properties of χ(mt, r, 2) MOB is ± MOB is LC No No No roots <0 No m, m 1 vals No FFB is LC Yes chromatic polynomial in MOB and FFB χ(mt, r, 2) = 6r 21r 2 + 7r r 4 14r 5 + r 7 = 84[r] [r] [r] [r] 6 + [r] 7 JM Møller (U of Copenhagen) Chromatic polynomials May 31, / 29

22 Are the simplicial Stirling numbers LC? Problem 2: Are the simplicial Stirling numbers j S(K, j, d), chr(k, d) j F 0 (K ) LC for fixed K and d? (Only property that might generalize!) log 10 S(S13,56 3, j, d) d = 3 d = 2 d = j chr(k, 1) = 10 chr(k, 2) = 3 chr(k, 3) = 2 JM Møller (U of Copenhagen) Chromatic polynomials May 31, / 29

23 Topology meets combinatorics Theorem 2.12 (Equivalent conditions for colorability) K admits an (r, d)-coloring There exists a lift such that Davis Januszkiewicz space r {}}{ BU(d) BU(d) DJ(K ) BU(1) BU(1) }{{} F 0 (K ) is homotopy commutative χ(k, r, d) > 0 λ 1 λ 1 BU λ d λ d JM Møller (U of Copenhagen) Chromatic polynomials May 31, / 29

24 The d-chromatic lattice T L(K, 2) T L(K, 2), π(t ) = 2 T L(K, 2), π(t ) = 1 Definition 3.1 The d-chromatic lattice, L(K, d), is the partially ordered set of monochrome subsets of F d (K ) of the form M d (col) = {σ F d (K ) col(σ) = 1} F d (K ) for some map col: F 0 (K ) {1,..., F 0 (K ) }. L(K, d) is a finite lattice with 0 = and 1 = F d (K ) µ is the Möbius function of L(K, d) π(t ) is the number of connected components of T L(K, d) JM Møller (U of Copenhagen) Chromatic polynomials May 31, / 29

25 Simplicial Stirling numbers and the chromatic lattice Theorem 3.2 (Relating simplicial and usual Stirling numbers) χ(k, r, d) = π(t ) µ( 0, T )r S(K, j, d) = T L(K,d) T L(K,d) µ( 0, T )S 2 ( π(t ), j) Dehn Sommerville relations for simplicial Stirling numbers of manifold? L(K, d) is graded for d = 1 but not for d > 1. Theorem 3.3 The reduced Euler characteristic of the open interval ( 0, 1) in L(K, d) is F 0 (K ) j=chr(k,d) ( 1) j 1 (j 1)!S(K, j, d) JM Møller (U of Copenhagen) Chromatic polynomials May 31, / 29

26 Integer sequences of Euler characteristics The reduced Euler characteristics of L(D[m], d)( 0, 1) for m d = 2, 3, 4,... are d = 1 : 2, 6, 24, 120, 720, 5040, 40320, ,... d = 2 : 3, 6, 0, 90, 630, 2520, 0, , ,... d = 3 : 4, 10, 20, 70, 560, 4200, 25200, ,... d = 4 : 5, 15, 35, 70, 0, 2100, 23100, , ,... d = 5 : 6, 21, 56, 126, 252, 924, 11088, ,... d = 6 : 7, 28, 84, 210, 462, 924, 0, 42042, ,... d = 7 : 8, 36, 120, 330, 792, 1716, 3432, 12870,... d = 8 : 9, 45, 165, 495, 1287, 3003, 6435, 12870, 0,... The first sequence is the sequence ( 1) m 1 (m 1)!. The second sequence is A from The On-Line Encyclopedia of Integer Sequences (OES). The remaining 6 sequences don t match any sequences of the OES. JM Møller (U of Copenhagen) Chromatic polynomials May 31, / 29

27 Weighted colorings Let w : F 0 (K ) N be a weight function on the vertices. The weight of a simplex σ K is the sum w(σ) = v σ w(v) of the weights of its vertices. (Special case: w = 1.) Definition 4.1 (Weighted (r, d)-coloring of K ) A (r, w d)-coloring of K is a function col: F 0 (K ) {1, 2,..., r} such that col(σ) = 1 = w(σ) d for all simplices σ K. Definition 4.2 (Weighted s-chromatic number of K ) The weighted d-chromatic number of K, chr(k, w d), is the minimal r so that K admits an (r, w d)-coloring. JM Møller (U of Copenhagen) Chromatic polynomials May 31, / 29

28 Weighted chromatic polynomials Definition 4.3 (Weighted simplicial Stirling numbers) S(K, j, w d) is the number of partitions of F 0 (K ) with j classes containing only simplices σ K of weight w(σ) d. Theorem 4.4 (Weighted d-chromatic polynomial) The number of weighted (r, d)-colorings of K is χ(k, r, w d) = F 0 (K ) j=chr(k,w d) S(K, j, w d)[r] j Problem 2: Are the weighted simplicial Stirling numbers j S(K, j, w d), chr(k, w d) j F 0 (K ) LC for fixed K, w, and d? JM Møller (U of Copenhagen) Chromatic polynomials May 31, / 29

29 Colorings as simplicial maps Definition 4.5 An (r, d)-coloring of K is a simplicial map col: K D[r] such that dim{σ K col(σ) = j} < d for 1 j r Definition 4.6 An (L, d)-coloring of K is a simplicial map col: K L such that dim col 1 (v) < d for all vertices v in L. JM Møller (U of Copenhagen) Chromatic polynomials May 31, / 29